Dynamics of a Non-Neutral Electron Plasma in Hall ... - MTU Aerospace
42nd AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit 9 - 12 July 2006, Sacramento, California
AIAA 2006-5173
Mobility Studies of a Pure Electron Plasma in Hall Thruster Fields Emily C. Fossum,* Lyon B. King,† and Jason Makela‡ Michigan Technological University, Houghton, MI, 49931, U.S.A An electron trapping apparatus was constructed in order to study electron dynamics in the defining electric and magnetic fields of a Hall-effect thruster. The approach presented here decouples the cross-field mobility from plasma effects by conducting measurements on a pure electron plasma in a highly controlled environment. Dielectric walls are removed completely eliminating all wall effects; thus, electrons are confined solely by a radial magnetic field and a crossed, independently-controlled, axial electric field that induces the closed-drift azimuthal Hall current. Electron trajectories and cross-field mobility were examined in response to electric and magnetic field strength and background neutral density. Without wall effects or neutral plasma effects mobility is presumed to follow the classical mobility model. In the present research, measurement techniques are investigated, and results are verified against the classical model. Preliminary findings suggest that that the apparatus and techniques used will be valid for mobility studies in more complex field environments.
Nomenclature Aa Ap Br B B bˆ 0
E Ez Id Ia Ip Ja Je Jez Jp me , ez
ne
ne
q rLe rLi ueB
0
= = = = = = = = = = = = = = = = = = = = = = = = =
anode surface area probe surface area radial magnetic field magnetic field vector magnitude of B unit vector in the B-field direction permittivity of free space electric field vector axial electric field mobility drift current anode current probe current current density at the anode Hall current density cross-field electron current density, current density at the probe electron mass magnetic moment of gyrating particle cross-field electron mobility electron number density electron-neutral collision frequency particle charge electron Larmor radius ion Larmor radius electron bounce velocity
*
GRA, Mechanical Engineering, [email protected] Associate Professor, Mechanical Engineering, [email protected] ‡ GRA, Mechanical Engineering, [email protected] 1 American Institute of Aeronautics and Astronautics †
Copyright © 2006 by Emily C. Fossum. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.
ue uez ve eff ce B H
T
= = = = = = = = =
azimuthal electron velocity cross-field electron velocity electron perpendicular velocity electric potential effective trap potential electron cyclotron frequency bounce frequency magnetron frequency electron Hall parameter
I.
Introduction
HE defining characteristic of Hall thrusters is the crossed axial electric and radial magnetic fields. The criteria of the E- and B-fields are such that the electron gyro radius is small compared with apparatus dimensions while the gyro radius and mean free path for ions are larger than apparatus dimensions; these criteria are necessary so that ions are only affected by the electric field, whereas the electron trajectories are controlled by both electric and magnetic fields. The purpose of the radial magnetic field in a Hall thruster is to impede electron flow toward the anode, which maintains the axial electric field necessary to accelerate large, unmagnetized propellant ions. The crossed E and B fields induce the confining ExB electron drift which holds electrons in azimuthal orbits around the discharge channel annulus. Electrons slowly diffuse across the radial B-field lines, gaining energy from the E-field and dissipating this energy through wall collisions and electron-neutral collisions, which ionize propellant neutrals. This cross-field electron transport is obviously necessary to sustain thruster discharge; however, mobility takes energy away from the accelerating region, which, in excess, has negative effect on thruster efficiency. In past experiments the cross-field electron mobility has been found to be much larger than the classical collisional diffusion model, which states that electrons cross field lines by electron-neutral collisions alone1,2. The anomalous electron mobility was first observed in Hall thruster fields by Janes and Lowder1 and later supported by Meezan et al2, who observed mobility up to 1,000 times greater than predicted by classical theory. Janes and Lowder predicted this departure to be caused in part by azimuthal fluctuations in plasma density, and thus electric field, creating a secondary drift term in the axial direction. Plasma fluctuations have since been characterized and are now well documented3 leading to Meezen’s work supporting Janes and Lowder’s initial hypothesis. Meezen provided the first quantified measure of mobility as it varies with axial position in the discharge channel, finding the greatest departure from the classical model at the location where plasma fluctuations were shown to exist. Others have hypothesized that the dielectric wall interactions play a significant role in electron transport, and attempts have been made to quantify and model the near wall region of the discharge channel4,5. To support the significance of wall interactions, King6, through a comprehensive study of forces in the discharge channel, described the necessity of a negative sheath on the outer dielectric wall in order to close the azimuthal ExB drift, expressing a radial E-field force ten times that of the magnetic mirror force acting on electrons at the outer wall. Without this negative sheath, which is responsible for the confining electric field, the dominant electron motion is toward the walls, resulting in multiple wall collisions in a single azimuthal orbit and a net current to the walls until this negative sheath is formed and sustained. The approach used in this investigation removes such plasma fluctuations and wall interactions by examining the electron dynamics of a pure electron plasma in a Hall thruster’s defining fields with an independently-controlled electric field in vacuum. Mobility studies were performed in a highly controlled environment disconnected from the coupling ion and neutral plasma effects that usually control and are controlled by the field environment. Also absent were the dielectric walls and consequently the sheath that exists adjacent to the dielectric surface. The properties of the sheath and pre-sheath region are complex and not easily defined. Removing these complexities allows the electric field to be known at all points within the apparatus allowing for greatly simplified diagnostics. While such an approach has not been documented in Hall thruster studies, an examination of non-neutral plasma has proven to be useful for numerous types of charged particle transport studies,7,8 examples being the Electron Diffusion Gauge (EDG) experiments of Chao and Davidson,9 the measurement of neo-classic mobility by Robertson,10,11,12 and numerous other related work involving positrons and ions.13 Without plasma fluctuations or wall-interactions, mobility in such Hall thruster fields can be studied in its purest sense and a comparison with classical and experimental mobility can be made.
2 American Institute of Aeronautics and Astronautics
II.
Mobility Concepts
In a Hall thruster the magnitude of the B-field is adjusted such that the electron Larmor radius is much less than the thruster size, while the ion Larmor radius is much greater. The net effect is that the electron trajectories are controlled by both the magnetic and electric fields, while the ion motion is affected only by the electric field, and the resulting motions can be uncoupled. From this, it follows that the predominant electron motion is the azimuthal ExB drift, since electrons are allowed to gyrate around B-field lines within the trap dimensions, and also gyrate around the discharge channel annulus, many times before undergoing a collision. In other words, the criteria in a Hall thruster discharge channel is such that the characteristic lengths and frequencies scale as follows: rLi>>rchannel>>rLe and ce> > ne. Classical cross-field electron mobility states that in a collisionless environment electrons would be indefinitely confined in azimuthal orbits. Collisions provide the only mechanism responsible for motion across B-field lines, through electron-neutral or electron-wall collisions, which allow an electron to “jump” to a new field line with a step length on the order of the Larmor radius. Therefore, increasing B slows cross-field diffusion by decreasing the Larmor radius and thus the step length. A detailed description of how this dependence comes about can be found in Chen14. Cross-field mobility is defined as the constant of proportionality between the cross-field velocity of electrons, uez, and the axial electric field, Ez:
µ ez ≡
uez . Ez
(1)
An analysis of mobility is presented here based on the assumption of a large Hall parameter, that is ce/ ne >> 1, a case that is required for Hall thruster operation. (In the situation that ce/ ne rtrap>>rLe and ce> > ne. The trap dimensions are on the order of 4 times larger than a typical 1.5 kW Hall thruster, with the outer diameter of the confining volume equal to 420 mm and a channel width of 100 mm. This allows us to operate in a wider range of B-field conditions, particularly low B-field, and still maintain scaling parameters. The magnetic field was produced by an inner and an outer coil inducing a field through iron magnet poles with the capability of producing a range of B-field magnitudes from ~30 Gauss to ~300 Gauss at the center of the confining volume. Much larger fields are possible with the present apparatus, however available power supplies limited testing to regions where B > ne. Cyclotron frequency (qB/me) is typically on the order of 10 -10 s for the field conditions presented. Magnetron frequency, described as the number of cycles around the entire trap annulus per unit of time, is determined from ue and an average trap radius ( =ue /2 rave). Magnetron frequency is on the order of 105-106 s-1 scaling inversely with magnetic field. A bounce frequency is described as the frequency at which an electron moves 8 American Institute of Aeronautics and Astronautics
back and forth radially within the trap as it is reflected by the net confining forces at the trap edges. Electrons are created by the ionizing beam uniformly across the width of the confinement volume along a magnetic field line with very little perpendicular energy16,17. If an electron is born at the center of the trap potential it will only have thermal energy (0.1 – 1 eV) parallel to the field line and will have very little radial travel. However, if an electron is born near the edge of the trap it will gain radial energy from the effective trap potential as it “rolls” toward the center of the trap volume and will continue to “bounce” between the inner and outer radii of the trap. Bounce frequency was determined numerically from the total time found for the electron to cross the channel width. For typical field conditions the bounce frequency was found to be on the order of 107 s-1 and did not vary greatly with varied field conditions. This frequency was also found to vary only minimally with the electron’s radial starting position, indicating that this motion can be approximated by a simple harmonic oscillator. The bounce frequency falls below electron cyclotron frequency but above magnetron frequency, which indicates a radial bounce as the electron traverses azimuthally. This motion resembles the characteristic electron motion described by King6. Since mobility is governed by electron-neutral collisions and since the electron-neutral collision frequency scales with electron velocity, it is important to determine which component of electron motion dictates the collision frequency. The characteristic electron trap motions are used in determining an effective momentum transfer collision frequency, as collision frequency can vary greatly with electron temperatures below 5 electron volts18. Although in general the characteristic frequencies scale as ce> B > within the operating regimes presented, the characteristic velocities corresponding to each of these frequencies (u , ueB and ue , respectively) do not necessarily scale as simply. To determine an order of magnitude upper bound for collision frequency, we identify the highest electron velocity; thus we consider the condition where B is the lowest such that ue is the greatest. At this point, the characteristic velocities scale as ue > ueB > u . Based on an effective collision frequency model for krypton given by Baille18, the effective collision frequency is on the order of 106 s-1 for operating pressures on the order of 5x10-6 torr. At the highest B fields investigated ue drops by an order of magnitude which changes the scaling to ueB> ue > u . In this case, had the effective collision frequency been determined based on ue , collision frequency would have been underestimated by more than three orders of magnitude. At large values of B the actual collision frequency is higher due to the bounce motion which then becomes the dominant motion of electrons. The effective velocity in this regime, ueB, is orders of magnitude greater than the azimuthal velocity. The bounce motion then gives rise to a lower bound for collision frequency of 6-8x105 s-1.
C. Experimental Determination of Mobility The cross-field mobility was evaluated experimentally by combining a measurement of the azimuthal Hall current with the axial (anode) current. The transverse mobility, ez, is related to the electron density, ne, the axial current flux, Jez, and the axial E-field, Ez, by Jez=qne ezEz. Thus, if the anode collects electron current, Ia, over an area Aa, the mobility can be expressed as µ ez =
Ja . qne E z
(10)
Because we know the axial field at every point in the thruster we only need to obtain a measure of electron density in order to experimentally quantify mobility; this is obtained by measuring the azimuthal Hall current with an in-situ probe. The current density incident on the probe is Jp, where J p = qne u eθ , and thus electron density can be determined, since ue = Ez/Br , where both Ez and Br are known from applied field conditions. Rearranged, this gives
ne =
J p Br qE z
.
(11)
By substitution, equation (10) simply becomes a ratio of the current density at the anode to the current density measured by the probe scaling with 1/Br: J (12) µ ez = a . Br J p This method has the benefit that the mobility is indicated by the current density ratio, rather than the absolute values of each current. This uncouples the effect of varying electron emission or pair production rate on the measured mobility. This experimentally determined mobility can then be compared quantitatively with the classical model using Eqn. 3. Also as described in the previous section, the probe measurement was used to monitor ne in order to ensure experiments are conducted within the space charge limitation. 9 American Institute of Aeronautics and Astronautics
For this in-situ measurement, a 2.36-mm-diameter probe was positioned 180 degrees opposite the ionizing electron beam, such that the probe does not collect current due to the high energy primaries but collects only lower energy, secondary electrons confined within the trap. The probe was biased 3V above local potential of the trap to be slightly attracting. This produced a negligible perturbation of the electric field of the trap and ensured that the probe was not repelling electron current should there have been error in field models or probe positioning. The probe was encased in an alumina sheath where the exposed collection area of the probe was aligned orthogonally to the azimuthal Hall current, as to collect only the directed flow of electrons. For typical field configurations to be investigated with this trap, namely Br~50-200 G and Ez~1x104 V/m, an electron cloud with density ~5x1010 m-3 would create a probe current on the order of tens of nano-Amps. During the course of experiments electron density was monitored and controlled to keep the electron density well below the space charge limit. All testing was performed in the Ion Space Propulsion Lab (Isp Lab) at Michigan Technological University. The testing facility consists of a 2-m-diameter, 4-m-long cylindrical vacuum chamber. Rough pumping is accomplished through a two-stage mechanical pump, capable of 400cfm. High vacuum is achieved through the use of three turbomolecular pumps with a combined throughput of 6,000 liters per second providing a base pressure below 10-6 Torr. Krypton gas was introduced to vary the base pressure from 10-6 to 10-4 Torr.
IV.
Results and Analysis
From Eqn. 12 it follows that experimental cross-field mobility can be found solely by a current density ratio between the anode and an in-situ probe measurement, since we know Br from applied fields. Currents were measured at the anode and the probe using a picoammeter and a digital source meter, respectively, in order to inspect mobility with changes in B-field magnitude, E-field magnitude and pressure (or equivalently, number density). The variation in cross-field mobility as a function of magnetic field was determined by monitoring the probe and anode current as the radial magnetic field was varied from 0.003 T to 0.018 T (30-180 Gauss) under constant facility vacuum pressure. The results are shown in Figure 7Error! Reference source not found. with a solid line representing classical mobility. Colors indicate several sweeps done with all conditions held constant, with the exception of slight variations between sweeps in base pressure due to time lapse between data acquisition. For low magnetic fields, experimental and classical mobility was found to agree with only slight departures at fields above 0.01 T (100 Gauss).
Cross field mobility,
ez
(m2/(V-s))
1.00E+01
1.00E+00
1.00E-01
1.00E-02 0.001
0.01
0.1
Radial Magnetic Field, Br (Tesla)
Figure 7. Cross-field mobility as it varies with radial magnetic field. Solid line represents classical mobility. Classical mobility depends on collision frequency, cyclotron frequency, and radial magnetic field (Eqn. 3). The radial magnetic field and cyclotron frequency can be easily determined from applied fields. However, as described previously, determining an effective collision frequency is not entirely straightforward. Effective collision frequency was determined from a model presented by Baille et al18 that gives collision frequency as a function of 10 American Institute of Aeronautics and Astronautics
electron temperature (or velocity) and number density since the momentum-transfer collision frequency has been found to vary greatly18 with electron energies in the range of 0.1 to 5 eV. Therefore, a true representative electron velocity is needed in order to calculate an accurate collision frequency. Figure 8 shows characteristic electron velocities as they vary with B-field. Azimuthal velocity is simply ue = Ez/Br where both Ez and Br are known from applied fields. Perpendicular velocity was calculated from the ejected electron energy (0.25 eV- 1eV corresponding to 2-4x104 m/s). Bounce velocity was determined numerically and was found to be on the order of 3-5 eV or 79x105 m/s.
Azimuthal velocity, ue Range of bounce velocity, uB
Range of perpendicular velocity, u
Figure 8. Analysis of electron velocity as it varies with Br (constant Ez) It is apparent that for low B-fields the azimuthal velocity is on the same order as the bounce velocity so either could be taken as the dominant electron motion; however, with increased B, the bounce velocity becomes the dominant motion as ue drops by more than an order of magnitude. The motion can be described as an electron “rattling” back and forth radially through the trap as it slowly traverses azimuthally. Therefore in the general determination of collision frequency the true electron velocity must be selected as the maximum of bounce velocity and azimuthal velocity. Velocity perpendicular to the magnetic field, u , is taken as the electron thermal velocity and is well below both the bounce and azimuthal velocities. Using this method for determining collision frequency, classical mobility was plotted in Figure 7Figure . There is visibly a region where ue is the dominant electron motion and a slight knee when the bounce velocity becomes the dominant electron motion. This method along with Baille’s model was used in all subsequent theoretical determinations of cross-field mobility. The variation of mobility with collision frequency was explored by bleeding background gas into the vacuum chamber. Krypton was introduced to the system to raise the base pressure from 9x10-7 to 1x10-4, and the observed cross-field mobility is shown in Figure 9. Colors indicate sweeps taken at the magnetic field magnitudes shown. Classical mobility is shown as solid lines in corresponding color. Upon first inspection, the trends exhibited by the observed mobility are reasonable; mobility scales with increasing neutral density and decreases with increasing magnetic field. However, it is recognized that classical mobility scales linearly with neutral density, by realizing that collision frequency and neutral density are linearly related and µez ∝ ν ne . Although the general trends of the observed mobility are understandable, the observed cross-field mobility was found to scale with nn1/2 through a power-law fit, rather than scaling linearly with nn as would be expected by the classical model. An explanation for this odd departure is still absent, as the authors plan to investigate this more thoroughly in future experiments.
11 American Institute of Aeronautics and Astronautics
1.00E+00
Cross field mobility,
ez
(m2/(V-s))
1.00E+01
1.00E-01
1.00E-02
1.00E-03 1E+16
46 G 87 G 128 G 1E+17
1E+18
1E+19
-3
Neutral Density, nn (m )
Figure 9. Cross-field mobility as it varies with neutral density for Kr. Solid lines in corresponding color indicate classical mobility Cross-field mobility was observed with variations in electric field over an order of magnitude, from ~ 2x103 to ~1x104 V/m and shown in Figure 10. Classical cross-field mobility is not expected to vary directly with electric field. However, an analysis of electron velocities in Figure 11 shows that in these field conditions the electron motion starts in a bounce dominated regime and then enters into a ue dominated regime where, ue scales linearly with Ez. Superimposing these two velocities gives the effective electron velocity over the electric field conditions considered. It follows, then, that collision frequency also varies over these field configurations translating to the non-constant classical mobility seen in Figure 10. Again there is good agreement between observed and classical mobility.
Cross field mobility,
ez
(m2/(V-s))
1.00E+00
1.00E-01
1.00E-02 1.00E+00 2.00E+03 4.00E+03 6.00E+03 8.00E+03 1.00E+04 1.20E+04 Axial Electric Field, Ez (Volts/meter)
Figure 10. Cross-field mobility as it varies with axial electric field. Solid lines indicate an upper and lower bound for classical mobility
12 American Institute of Aeronautics and Astronautics
Azimuthal velocity, ue
Range of bounce velocity, uB Range of perpendicular velocity, u
Figure 11. Analysis of electron velocity as it varies with Ez (constant Br)
Over the conditions examined an electron density was observed to ensure all experiments were performed within the space charge limit. Figure 12 shows a plot of the electron number density as a function of B as measured by the electrostatic probe, with a maximum of 4.5x1010 m-3 which is well within our space charge limit. 5.00E+10
-3
Electron Number Density, ne (m )
4.50E+10 4.00E+10 3.50E+10 3.00E+10 2.50E+10 2.00E+10 1.50E+10 1.00E+10 5.00E+09 0.00E+00 0
0.005
0.01
0.015
0.02
Radial Magnetic Field, Br (Tesla)
Figure 12. Electron number density as it varies with radial magnetic field. The characteristic of the electron density profile as it varies with B is particularly interesting. At low B-fields, many high energy emitted electrons are lost because their Larmor radius is larger than trap dimensions. However, as B field is increased, more primary electrons are confined by the B-field and allowed to create electron-ion pairs. This trend is expected to increase to a saturation value where a maximum number of ionizing primaries are sent through the trap. However, instead of saturating, the electron density, as measured by the probe, sharply declined at fields above ~100 Gauss. This unusual characteristic is thought to be due to an artifact of the trap loading mechanism (see Figure 13Error! Reference source not found.). When the trap is loaded at low B-fields the Larmor radius of the primary ionizing electrons is on the order of the trap dimensions, creating electron-ion pairs 13 American Institute of Aeronautics and Astronautics
over the entire confinement volume of the trap. However, at high B-fields the primary ionizing electrons have a Larmor radius that is a fraction of the trap volume so electrons are only created in a small slice of the confinement volume, which is not seen directly by the probe. These two loading regimes have an obvious effect on the current incident at the probe, which could be mistaken as changes in mobility. In order to address this matter authors indend to incorporate an emitting filament that extends over the entire height of the confinement volume such that electrons are created uniformly over the confinement volume in all B-field conditions.
High B-field trajectory
Ionizing volume for high B-field
Ionizing volume for low B-field Low B-field trajectory
Figure 13.
Schematic of trap loading mechanism.
V.
Conclusions
Concepts have been explored for using a non-neutral plasma in order to study mobility in Hall thrusters. This paper presents a description of an apparatus that can be used to investigate electron mobility in Hall thruster fields absent from plasma effects and wall effects. The approach presented introduces a new method for Hall thruster research by decoupling the cross-field mobility from complicating plasma fluctuations and turbulence by conducting measurements on a pure electron plasma in a highly controlled environment. The first stage of this research was to measure mobility in simplified fields and to validate it against the classical mobility model. This method has been verified by the clear agreement between observed and classical cross-field mobilities with variations in magnetic and electric fields and neutral density. However, it was discovered from this research that an effective electron velocity is of utmost importance when calculating classical mobility. An accurate determination of classical mobility depends on an effective value of collision frequency, which can only be established through an analysis of dominant electron motions in the trap apparatus. The preliminary findings suggest that this method is suitable for Hall thruster research and will continue to be valid in more complex field environments. The ability to decouple electron motion from plasma effects and control the electric field externally gives rise to a wealth of experiments involving optimized magnetic fields or externally applied electrostatic perturbations.
Acknowledgments The authors wish to acknowledge valuable discussions with and the assistance of Dean Massey and Alex Kieckhafer in this research. The authors would also like to acknowledge Master Machinist, Marty Toth, for all of the hard work and long hours required in the fabrication of the electron trapping apparatus. This work was supported by the National Science Foundation. 14 American Institute of Aeronautics and Astronautics
References 1
Janes, G.S., and Lowder, R.S., “Anomalous Electron Diffusion and Ion Acceleration in a Low-Density Plasma,” Physics of Fluids, Vol. 9, No. 6, (1966), pp. 1115-1123. 2
Meezan, N.B., Hargus, W.A. Jr., and Cappelli, M.A., “Anomalous Electron Mobility in a Coaxial Hall Discharge Plasma,” Physical Review E., Vol. 63, No. 2, (2001) pp. 026410-1-7 3
G. Guerrini and C. Michaut, “Characterization of high frequency oscillations in a small Hall-type thruster,” Physics of Plasmas, Vol. 6, (1999) pp. 343.
4
M. Keidar et al, “Plasma flow and plasma-wall transition in Hall Thruster Channel,” Physics of Plasmas, Vol. 8, pp. 5315.
5
J.P. Boeuf and L. Garrigues, “Low Frequency Oscillations in a Stationary Plasma Thruster,” Journal of Applied Physics, Vol. 84, pp. 3541.
6 King, L. B., “A (Re-)Examination of Electron Motion in Hall Thruster Fields”, 29th International Electric Propulsion Conference, IEPC-2005-258. 7
Malmberg, J.H., Driscoll, C.F., Beck, B., Eggleston, D.L., Fajans, J., Fine, K., Huang, X.-P., and Hyatt, A.W., in Non-Neutral Plasma Physics, edited by C.W. Roberson and C.F. Driscoll, AIP Conference Proc. Vol. 175, No. 28, 1988. 8
deGrassie, J.S., and Malmberg, J.H., “Waves and transport in the pure electron plasma,” Physics of Fluids, Vol. 23, 63 (1980).
9
Chao, E.H., Davidson, R.C., Paul, S.F., and Morrison, K.A., “Effects of background gas pressure on the dynamics of a nonneutral electron plasma confined in a Malmberg-Penning trap,” Physics of Plasmas, Vol. 7, No. 3, March 2000, pp. 831-838.
10
Robertson, S., and Walch, B., “Electron confinement in an annular Penning trap,” Physics of Plasmas, Vol. 7, No. 6, June 2000, pp. 2340-2347. 11 Espejo, J., Quraishi, Q., and Robertson, S., “Experimental measurement of neoclassic mobility in an annular MalmbergPenning trap,” Physical Review Letters, Vol. 84, No. 24, 12 June 2000, pp. 5520-5523. 12 Robertson, S., Espejo, J., Kline, J., Quraishi, Q, Triplett, M., and Walch, B., “Neoclassical effects in the annular Penning trap,” Physics of Plasmas, Vol. 8, No. 5, May 2001, pp. 1863-1869. 13
for an extensive review of related non-neutral plasma trapping research see Bollinger, J.J., Spencer, R.L., and Davidson, R.C., eds., Non-neutral Plasma Physics, AIP Conference Proc. 498, Aug. 1999 and Dubin, D.H.E., and Schneider, D., eds., Trapped Charged Particles and Fundamental Physics, AIP Conference Proc. 457, Aug. 1998. 14
Chen, F. F., Introduction to Plasma Physics and Controlled Fusion, 2nd ed., Plenum Press, New York, 1984, Chap. 5.
15
Linnell, J. A. and Gallimore, A. D., “Internal Plasma Structure Measurements Using Xenon and Krypton Propellant,” International Electric Propulsion Conference, IEPC2005-024.
16
Peterson, W. K., Beaty, E. C., and Opal, C. B., “Measurements of Energy and Angular Distributions of Secondary Electrons Produced in Electron-Impact Ionization of Helium,” Physical Review A, Vol. 5, No. 2, Feb. 1972, pp. 712-723
17
Grissom, J. T., Compton, R. N. and Garrett, W. R., “Slow Electrons from Electron-Impact Ionization of He, Ne, and Ar,” Physical Review A, Vol. 6, No. 3, Sept. 1972, pp. 977-987 18
Baille, P., et al. “Effective Collision Frequency of Electrons in Noble Gases,” Journal of Physics B: Atomic and Molecular Physics, Vol. 14, (1981), pp. 1485-1495.
15 American Institute of Aeronautics and Astronautics
AIAA 2006-5173
Mobility Studies of a Pure Electron Plasma in Hall Thruster Fields Emily C. Fossum,* Lyon B. King,† and Jason Makela‡ Michigan Technological University, Houghton, MI, 49931, U.S.A An electron trapping apparatus was constructed in order to study electron dynamics in the defining electric and magnetic fields of a Hall-effect thruster. The approach presented here decouples the cross-field mobility from plasma effects by conducting measurements on a pure electron plasma in a highly controlled environment. Dielectric walls are removed completely eliminating all wall effects; thus, electrons are confined solely by a radial magnetic field and a crossed, independently-controlled, axial electric field that induces the closed-drift azimuthal Hall current. Electron trajectories and cross-field mobility were examined in response to electric and magnetic field strength and background neutral density. Without wall effects or neutral plasma effects mobility is presumed to follow the classical mobility model. In the present research, measurement techniques are investigated, and results are verified against the classical model. Preliminary findings suggest that that the apparatus and techniques used will be valid for mobility studies in more complex field environments.
Nomenclature Aa Ap Br B B bˆ 0
E Ez Id Ia Ip Ja Je Jez Jp me , ez
ne
ne
q rLe rLi ueB
0
= = = = = = = = = = = = = = = = = = = = = = = = =
anode surface area probe surface area radial magnetic field magnetic field vector magnitude of B unit vector in the B-field direction permittivity of free space electric field vector axial electric field mobility drift current anode current probe current current density at the anode Hall current density cross-field electron current density, current density at the probe electron mass magnetic moment of gyrating particle cross-field electron mobility electron number density electron-neutral collision frequency particle charge electron Larmor radius ion Larmor radius electron bounce velocity
*
GRA, Mechanical Engineering, [email protected] Associate Professor, Mechanical Engineering, [email protected] ‡ GRA, Mechanical Engineering, [email protected] 1 American Institute of Aeronautics and Astronautics †
Copyright © 2006 by Emily C. Fossum. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.
ue uez ve eff ce B H
T
= = = = = = = = =
azimuthal electron velocity cross-field electron velocity electron perpendicular velocity electric potential effective trap potential electron cyclotron frequency bounce frequency magnetron frequency electron Hall parameter
I.
Introduction
HE defining characteristic of Hall thrusters is the crossed axial electric and radial magnetic fields. The criteria of the E- and B-fields are such that the electron gyro radius is small compared with apparatus dimensions while the gyro radius and mean free path for ions are larger than apparatus dimensions; these criteria are necessary so that ions are only affected by the electric field, whereas the electron trajectories are controlled by both electric and magnetic fields. The purpose of the radial magnetic field in a Hall thruster is to impede electron flow toward the anode, which maintains the axial electric field necessary to accelerate large, unmagnetized propellant ions. The crossed E and B fields induce the confining ExB electron drift which holds electrons in azimuthal orbits around the discharge channel annulus. Electrons slowly diffuse across the radial B-field lines, gaining energy from the E-field and dissipating this energy through wall collisions and electron-neutral collisions, which ionize propellant neutrals. This cross-field electron transport is obviously necessary to sustain thruster discharge; however, mobility takes energy away from the accelerating region, which, in excess, has negative effect on thruster efficiency. In past experiments the cross-field electron mobility has been found to be much larger than the classical collisional diffusion model, which states that electrons cross field lines by electron-neutral collisions alone1,2. The anomalous electron mobility was first observed in Hall thruster fields by Janes and Lowder1 and later supported by Meezan et al2, who observed mobility up to 1,000 times greater than predicted by classical theory. Janes and Lowder predicted this departure to be caused in part by azimuthal fluctuations in plasma density, and thus electric field, creating a secondary drift term in the axial direction. Plasma fluctuations have since been characterized and are now well documented3 leading to Meezen’s work supporting Janes and Lowder’s initial hypothesis. Meezen provided the first quantified measure of mobility as it varies with axial position in the discharge channel, finding the greatest departure from the classical model at the location where plasma fluctuations were shown to exist. Others have hypothesized that the dielectric wall interactions play a significant role in electron transport, and attempts have been made to quantify and model the near wall region of the discharge channel4,5. To support the significance of wall interactions, King6, through a comprehensive study of forces in the discharge channel, described the necessity of a negative sheath on the outer dielectric wall in order to close the azimuthal ExB drift, expressing a radial E-field force ten times that of the magnetic mirror force acting on electrons at the outer wall. Without this negative sheath, which is responsible for the confining electric field, the dominant electron motion is toward the walls, resulting in multiple wall collisions in a single azimuthal orbit and a net current to the walls until this negative sheath is formed and sustained. The approach used in this investigation removes such plasma fluctuations and wall interactions by examining the electron dynamics of a pure electron plasma in a Hall thruster’s defining fields with an independently-controlled electric field in vacuum. Mobility studies were performed in a highly controlled environment disconnected from the coupling ion and neutral plasma effects that usually control and are controlled by the field environment. Also absent were the dielectric walls and consequently the sheath that exists adjacent to the dielectric surface. The properties of the sheath and pre-sheath region are complex and not easily defined. Removing these complexities allows the electric field to be known at all points within the apparatus allowing for greatly simplified diagnostics. While such an approach has not been documented in Hall thruster studies, an examination of non-neutral plasma has proven to be useful for numerous types of charged particle transport studies,7,8 examples being the Electron Diffusion Gauge (EDG) experiments of Chao and Davidson,9 the measurement of neo-classic mobility by Robertson,10,11,12 and numerous other related work involving positrons and ions.13 Without plasma fluctuations or wall-interactions, mobility in such Hall thruster fields can be studied in its purest sense and a comparison with classical and experimental mobility can be made.
2 American Institute of Aeronautics and Astronautics
II.
Mobility Concepts
In a Hall thruster the magnitude of the B-field is adjusted such that the electron Larmor radius is much less than the thruster size, while the ion Larmor radius is much greater. The net effect is that the electron trajectories are controlled by both the magnetic and electric fields, while the ion motion is affected only by the electric field, and the resulting motions can be uncoupled. From this, it follows that the predominant electron motion is the azimuthal ExB drift, since electrons are allowed to gyrate around B-field lines within the trap dimensions, and also gyrate around the discharge channel annulus, many times before undergoing a collision. In other words, the criteria in a Hall thruster discharge channel is such that the characteristic lengths and frequencies scale as follows: rLi>>rchannel>>rLe and ce> > ne. Classical cross-field electron mobility states that in a collisionless environment electrons would be indefinitely confined in azimuthal orbits. Collisions provide the only mechanism responsible for motion across B-field lines, through electron-neutral or electron-wall collisions, which allow an electron to “jump” to a new field line with a step length on the order of the Larmor radius. Therefore, increasing B slows cross-field diffusion by decreasing the Larmor radius and thus the step length. A detailed description of how this dependence comes about can be found in Chen14. Cross-field mobility is defined as the constant of proportionality between the cross-field velocity of electrons, uez, and the axial electric field, Ez:
µ ez ≡
uez . Ez
(1)
An analysis of mobility is presented here based on the assumption of a large Hall parameter, that is ce/ ne >> 1, a case that is required for Hall thruster operation. (In the situation that ce/ ne rtrap>>rLe and ce> > ne. The trap dimensions are on the order of 4 times larger than a typical 1.5 kW Hall thruster, with the outer diameter of the confining volume equal to 420 mm and a channel width of 100 mm. This allows us to operate in a wider range of B-field conditions, particularly low B-field, and still maintain scaling parameters. The magnetic field was produced by an inner and an outer coil inducing a field through iron magnet poles with the capability of producing a range of B-field magnitudes from ~30 Gauss to ~300 Gauss at the center of the confining volume. Much larger fields are possible with the present apparatus, however available power supplies limited testing to regions where B > ne. Cyclotron frequency (qB/me) is typically on the order of 10 -10 s for the field conditions presented. Magnetron frequency, described as the number of cycles around the entire trap annulus per unit of time, is determined from ue and an average trap radius ( =ue /2 rave). Magnetron frequency is on the order of 105-106 s-1 scaling inversely with magnetic field. A bounce frequency is described as the frequency at which an electron moves 8 American Institute of Aeronautics and Astronautics
back and forth radially within the trap as it is reflected by the net confining forces at the trap edges. Electrons are created by the ionizing beam uniformly across the width of the confinement volume along a magnetic field line with very little perpendicular energy16,17. If an electron is born at the center of the trap potential it will only have thermal energy (0.1 – 1 eV) parallel to the field line and will have very little radial travel. However, if an electron is born near the edge of the trap it will gain radial energy from the effective trap potential as it “rolls” toward the center of the trap volume and will continue to “bounce” between the inner and outer radii of the trap. Bounce frequency was determined numerically from the total time found for the electron to cross the channel width. For typical field conditions the bounce frequency was found to be on the order of 107 s-1 and did not vary greatly with varied field conditions. This frequency was also found to vary only minimally with the electron’s radial starting position, indicating that this motion can be approximated by a simple harmonic oscillator. The bounce frequency falls below electron cyclotron frequency but above magnetron frequency, which indicates a radial bounce as the electron traverses azimuthally. This motion resembles the characteristic electron motion described by King6. Since mobility is governed by electron-neutral collisions and since the electron-neutral collision frequency scales with electron velocity, it is important to determine which component of electron motion dictates the collision frequency. The characteristic electron trap motions are used in determining an effective momentum transfer collision frequency, as collision frequency can vary greatly with electron temperatures below 5 electron volts18. Although in general the characteristic frequencies scale as ce> B > within the operating regimes presented, the characteristic velocities corresponding to each of these frequencies (u , ueB and ue , respectively) do not necessarily scale as simply. To determine an order of magnitude upper bound for collision frequency, we identify the highest electron velocity; thus we consider the condition where B is the lowest such that ue is the greatest. At this point, the characteristic velocities scale as ue > ueB > u . Based on an effective collision frequency model for krypton given by Baille18, the effective collision frequency is on the order of 106 s-1 for operating pressures on the order of 5x10-6 torr. At the highest B fields investigated ue drops by an order of magnitude which changes the scaling to ueB> ue > u . In this case, had the effective collision frequency been determined based on ue , collision frequency would have been underestimated by more than three orders of magnitude. At large values of B the actual collision frequency is higher due to the bounce motion which then becomes the dominant motion of electrons. The effective velocity in this regime, ueB, is orders of magnitude greater than the azimuthal velocity. The bounce motion then gives rise to a lower bound for collision frequency of 6-8x105 s-1.
C. Experimental Determination of Mobility The cross-field mobility was evaluated experimentally by combining a measurement of the azimuthal Hall current with the axial (anode) current. The transverse mobility, ez, is related to the electron density, ne, the axial current flux, Jez, and the axial E-field, Ez, by Jez=qne ezEz. Thus, if the anode collects electron current, Ia, over an area Aa, the mobility can be expressed as µ ez =
Ja . qne E z
(10)
Because we know the axial field at every point in the thruster we only need to obtain a measure of electron density in order to experimentally quantify mobility; this is obtained by measuring the azimuthal Hall current with an in-situ probe. The current density incident on the probe is Jp, where J p = qne u eθ , and thus electron density can be determined, since ue = Ez/Br , where both Ez and Br are known from applied field conditions. Rearranged, this gives
ne =
J p Br qE z
.
(11)
By substitution, equation (10) simply becomes a ratio of the current density at the anode to the current density measured by the probe scaling with 1/Br: J (12) µ ez = a . Br J p This method has the benefit that the mobility is indicated by the current density ratio, rather than the absolute values of each current. This uncouples the effect of varying electron emission or pair production rate on the measured mobility. This experimentally determined mobility can then be compared quantitatively with the classical model using Eqn. 3. Also as described in the previous section, the probe measurement was used to monitor ne in order to ensure experiments are conducted within the space charge limitation. 9 American Institute of Aeronautics and Astronautics
For this in-situ measurement, a 2.36-mm-diameter probe was positioned 180 degrees opposite the ionizing electron beam, such that the probe does not collect current due to the high energy primaries but collects only lower energy, secondary electrons confined within the trap. The probe was biased 3V above local potential of the trap to be slightly attracting. This produced a negligible perturbation of the electric field of the trap and ensured that the probe was not repelling electron current should there have been error in field models or probe positioning. The probe was encased in an alumina sheath where the exposed collection area of the probe was aligned orthogonally to the azimuthal Hall current, as to collect only the directed flow of electrons. For typical field configurations to be investigated with this trap, namely Br~50-200 G and Ez~1x104 V/m, an electron cloud with density ~5x1010 m-3 would create a probe current on the order of tens of nano-Amps. During the course of experiments electron density was monitored and controlled to keep the electron density well below the space charge limit. All testing was performed in the Ion Space Propulsion Lab (Isp Lab) at Michigan Technological University. The testing facility consists of a 2-m-diameter, 4-m-long cylindrical vacuum chamber. Rough pumping is accomplished through a two-stage mechanical pump, capable of 400cfm. High vacuum is achieved through the use of three turbomolecular pumps with a combined throughput of 6,000 liters per second providing a base pressure below 10-6 Torr. Krypton gas was introduced to vary the base pressure from 10-6 to 10-4 Torr.
IV.
Results and Analysis
From Eqn. 12 it follows that experimental cross-field mobility can be found solely by a current density ratio between the anode and an in-situ probe measurement, since we know Br from applied fields. Currents were measured at the anode and the probe using a picoammeter and a digital source meter, respectively, in order to inspect mobility with changes in B-field magnitude, E-field magnitude and pressure (or equivalently, number density). The variation in cross-field mobility as a function of magnetic field was determined by monitoring the probe and anode current as the radial magnetic field was varied from 0.003 T to 0.018 T (30-180 Gauss) under constant facility vacuum pressure. The results are shown in Figure 7Error! Reference source not found. with a solid line representing classical mobility. Colors indicate several sweeps done with all conditions held constant, with the exception of slight variations between sweeps in base pressure due to time lapse between data acquisition. For low magnetic fields, experimental and classical mobility was found to agree with only slight departures at fields above 0.01 T (100 Gauss).
Cross field mobility,
ez
(m2/(V-s))
1.00E+01
1.00E+00
1.00E-01
1.00E-02 0.001
0.01
0.1
Radial Magnetic Field, Br (Tesla)
Figure 7. Cross-field mobility as it varies with radial magnetic field. Solid line represents classical mobility. Classical mobility depends on collision frequency, cyclotron frequency, and radial magnetic field (Eqn. 3). The radial magnetic field and cyclotron frequency can be easily determined from applied fields. However, as described previously, determining an effective collision frequency is not entirely straightforward. Effective collision frequency was determined from a model presented by Baille et al18 that gives collision frequency as a function of 10 American Institute of Aeronautics and Astronautics
electron temperature (or velocity) and number density since the momentum-transfer collision frequency has been found to vary greatly18 with electron energies in the range of 0.1 to 5 eV. Therefore, a true representative electron velocity is needed in order to calculate an accurate collision frequency. Figure 8 shows characteristic electron velocities as they vary with B-field. Azimuthal velocity is simply ue = Ez/Br where both Ez and Br are known from applied fields. Perpendicular velocity was calculated from the ejected electron energy (0.25 eV- 1eV corresponding to 2-4x104 m/s). Bounce velocity was determined numerically and was found to be on the order of 3-5 eV or 79x105 m/s.
Azimuthal velocity, ue Range of bounce velocity, uB
Range of perpendicular velocity, u
Figure 8. Analysis of electron velocity as it varies with Br (constant Ez) It is apparent that for low B-fields the azimuthal velocity is on the same order as the bounce velocity so either could be taken as the dominant electron motion; however, with increased B, the bounce velocity becomes the dominant motion as ue drops by more than an order of magnitude. The motion can be described as an electron “rattling” back and forth radially through the trap as it slowly traverses azimuthally. Therefore in the general determination of collision frequency the true electron velocity must be selected as the maximum of bounce velocity and azimuthal velocity. Velocity perpendicular to the magnetic field, u , is taken as the electron thermal velocity and is well below both the bounce and azimuthal velocities. Using this method for determining collision frequency, classical mobility was plotted in Figure 7Figure . There is visibly a region where ue is the dominant electron motion and a slight knee when the bounce velocity becomes the dominant electron motion. This method along with Baille’s model was used in all subsequent theoretical determinations of cross-field mobility. The variation of mobility with collision frequency was explored by bleeding background gas into the vacuum chamber. Krypton was introduced to the system to raise the base pressure from 9x10-7 to 1x10-4, and the observed cross-field mobility is shown in Figure 9. Colors indicate sweeps taken at the magnetic field magnitudes shown. Classical mobility is shown as solid lines in corresponding color. Upon first inspection, the trends exhibited by the observed mobility are reasonable; mobility scales with increasing neutral density and decreases with increasing magnetic field. However, it is recognized that classical mobility scales linearly with neutral density, by realizing that collision frequency and neutral density are linearly related and µez ∝ ν ne . Although the general trends of the observed mobility are understandable, the observed cross-field mobility was found to scale with nn1/2 through a power-law fit, rather than scaling linearly with nn as would be expected by the classical model. An explanation for this odd departure is still absent, as the authors plan to investigate this more thoroughly in future experiments.
11 American Institute of Aeronautics and Astronautics
1.00E+00
Cross field mobility,
ez
(m2/(V-s))
1.00E+01
1.00E-01
1.00E-02
1.00E-03 1E+16
46 G 87 G 128 G 1E+17
1E+18
1E+19
-3
Neutral Density, nn (m )
Figure 9. Cross-field mobility as it varies with neutral density for Kr. Solid lines in corresponding color indicate classical mobility Cross-field mobility was observed with variations in electric field over an order of magnitude, from ~ 2x103 to ~1x104 V/m and shown in Figure 10. Classical cross-field mobility is not expected to vary directly with electric field. However, an analysis of electron velocities in Figure 11 shows that in these field conditions the electron motion starts in a bounce dominated regime and then enters into a ue dominated regime where, ue scales linearly with Ez. Superimposing these two velocities gives the effective electron velocity over the electric field conditions considered. It follows, then, that collision frequency also varies over these field configurations translating to the non-constant classical mobility seen in Figure 10. Again there is good agreement between observed and classical mobility.
Cross field mobility,
ez
(m2/(V-s))
1.00E+00
1.00E-01
1.00E-02 1.00E+00 2.00E+03 4.00E+03 6.00E+03 8.00E+03 1.00E+04 1.20E+04 Axial Electric Field, Ez (Volts/meter)
Figure 10. Cross-field mobility as it varies with axial electric field. Solid lines indicate an upper and lower bound for classical mobility
12 American Institute of Aeronautics and Astronautics
Azimuthal velocity, ue
Range of bounce velocity, uB Range of perpendicular velocity, u
Figure 11. Analysis of electron velocity as it varies with Ez (constant Br)
Over the conditions examined an electron density was observed to ensure all experiments were performed within the space charge limit. Figure 12 shows a plot of the electron number density as a function of B as measured by the electrostatic probe, with a maximum of 4.5x1010 m-3 which is well within our space charge limit. 5.00E+10
-3
Electron Number Density, ne (m )
4.50E+10 4.00E+10 3.50E+10 3.00E+10 2.50E+10 2.00E+10 1.50E+10 1.00E+10 5.00E+09 0.00E+00 0
0.005
0.01
0.015
0.02
Radial Magnetic Field, Br (Tesla)
Figure 12. Electron number density as it varies with radial magnetic field. The characteristic of the electron density profile as it varies with B is particularly interesting. At low B-fields, many high energy emitted electrons are lost because their Larmor radius is larger than trap dimensions. However, as B field is increased, more primary electrons are confined by the B-field and allowed to create electron-ion pairs. This trend is expected to increase to a saturation value where a maximum number of ionizing primaries are sent through the trap. However, instead of saturating, the electron density, as measured by the probe, sharply declined at fields above ~100 Gauss. This unusual characteristic is thought to be due to an artifact of the trap loading mechanism (see Figure 13Error! Reference source not found.). When the trap is loaded at low B-fields the Larmor radius of the primary ionizing electrons is on the order of the trap dimensions, creating electron-ion pairs 13 American Institute of Aeronautics and Astronautics
over the entire confinement volume of the trap. However, at high B-fields the primary ionizing electrons have a Larmor radius that is a fraction of the trap volume so electrons are only created in a small slice of the confinement volume, which is not seen directly by the probe. These two loading regimes have an obvious effect on the current incident at the probe, which could be mistaken as changes in mobility. In order to address this matter authors indend to incorporate an emitting filament that extends over the entire height of the confinement volume such that electrons are created uniformly over the confinement volume in all B-field conditions.
High B-field trajectory
Ionizing volume for high B-field
Ionizing volume for low B-field Low B-field trajectory
Figure 13.
Schematic of trap loading mechanism.
V.
Conclusions
Concepts have been explored for using a non-neutral plasma in order to study mobility in Hall thrusters. This paper presents a description of an apparatus that can be used to investigate electron mobility in Hall thruster fields absent from plasma effects and wall effects. The approach presented introduces a new method for Hall thruster research by decoupling the cross-field mobility from complicating plasma fluctuations and turbulence by conducting measurements on a pure electron plasma in a highly controlled environment. The first stage of this research was to measure mobility in simplified fields and to validate it against the classical mobility model. This method has been verified by the clear agreement between observed and classical cross-field mobilities with variations in magnetic and electric fields and neutral density. However, it was discovered from this research that an effective electron velocity is of utmost importance when calculating classical mobility. An accurate determination of classical mobility depends on an effective value of collision frequency, which can only be established through an analysis of dominant electron motions in the trap apparatus. The preliminary findings suggest that this method is suitable for Hall thruster research and will continue to be valid in more complex field environments. The ability to decouple electron motion from plasma effects and control the electric field externally gives rise to a wealth of experiments involving optimized magnetic fields or externally applied electrostatic perturbations.
Acknowledgments The authors wish to acknowledge valuable discussions with and the assistance of Dean Massey and Alex Kieckhafer in this research. The authors would also like to acknowledge Master Machinist, Marty Toth, for all of the hard work and long hours required in the fabrication of the electron trapping apparatus. This work was supported by the National Science Foundation. 14 American Institute of Aeronautics and Astronautics
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Janes, G.S., and Lowder, R.S., “Anomalous Electron Diffusion and Ion Acceleration in a Low-Density Plasma,” Physics of Fluids, Vol. 9, No. 6, (1966), pp. 1115-1123. 2
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4
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5
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6 King, L. B., “A (Re-)Examination of Electron Motion in Hall Thruster Fields”, 29th International Electric Propulsion Conference, IEPC-2005-258. 7
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9
Chao, E.H., Davidson, R.C., Paul, S.F., and Morrison, K.A., “Effects of background gas pressure on the dynamics of a nonneutral electron plasma confined in a Malmberg-Penning trap,” Physics of Plasmas, Vol. 7, No. 3, March 2000, pp. 831-838.
10
Robertson, S., and Walch, B., “Electron confinement in an annular Penning trap,” Physics of Plasmas, Vol. 7, No. 6, June 2000, pp. 2340-2347. 11 Espejo, J., Quraishi, Q., and Robertson, S., “Experimental measurement of neoclassic mobility in an annular MalmbergPenning trap,” Physical Review Letters, Vol. 84, No. 24, 12 June 2000, pp. 5520-5523. 12 Robertson, S., Espejo, J., Kline, J., Quraishi, Q, Triplett, M., and Walch, B., “Neoclassical effects in the annular Penning trap,” Physics of Plasmas, Vol. 8, No. 5, May 2001, pp. 1863-1869. 13
for an extensive review of related non-neutral plasma trapping research see Bollinger, J.J., Spencer, R.L., and Davidson, R.C., eds., Non-neutral Plasma Physics, AIP Conference Proc. 498, Aug. 1999 and Dubin, D.H.E., and Schneider, D., eds., Trapped Charged Particles and Fundamental Physics, AIP Conference Proc. 457, Aug. 1998. 14
Chen, F. F., Introduction to Plasma Physics and Controlled Fusion, 2nd ed., Plenum Press, New York, 1984, Chap. 5.
15
Linnell, J. A. and Gallimore, A. D., “Internal Plasma Structure Measurements Using Xenon and Krypton Propellant,” International Electric Propulsion Conference, IEPC2005-024.
16
Peterson, W. K., Beaty, E. C., and Opal, C. B., “Measurements of Energy and Angular Distributions of Secondary Electrons Produced in Electron-Impact Ionization of Helium,” Physical Review A, Vol. 5, No. 2, Feb. 1972, pp. 712-723
17
Grissom, J. T., Compton, R. N. and Garrett, W. R., “Slow Electrons from Electron-Impact Ionization of He, Ne, and Ar,” Physical Review A, Vol. 6, No. 3, Sept. 1972, pp. 977-987 18
Baille, P., et al. “Effective Collision Frequency of Electrons in Noble Gases,” Journal of Physics B: Atomic and Molecular Physics, Vol. 14, (1981), pp. 1485-1495.
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